34,905 research outputs found
Afterglow lightcurves, viewing angle and the jet structure of gamma-ray bursts
Gamma ray bursts are often modelled as jet-like outflows directed towards the
observer; the cone angle of the jet is then commonly inferred from the time at
which there is a steepening in the power-law decay of the afterglow. We
consider an alternative model in which the jet has a beam pattern where the
luminosity per unit solid angle (and perhaps also the initial Lorentz factor)
decreases smoothly away from the axis, rather than having a well-defined cone
angle within which the flow is uniform. We show that the break in the afterglow
light curve then occurs at a time that depends on the viewing angle. Instead of
implying a range of intrinsically different jets - some very narrow, and others
with similar power spread over a wider cone - the data on afterglow breaks
could be consistent with a standardized jet, viewed from different angles. We
discuss the implication of this model for the luminosity function.Comment: Corrected typo in Eq. 1
An obstacle problem for Tug-of-War games
We consider the obstacle problem for the infinity Laplace equation. Given a
Lipschitz boundary function and a Lipschitz obstacle we prove the existence and
uniqueness of a super infinity-harmonic function constrained to lie above the
obstacle which is infinity harmonic where it lies strictly above the obstacle.
Moreover, we show that this function is the limit of value functions of a game
we call obstacle tug-of-war
Shape maps for second order partial differential equations
We analyse the singularity formation of congruences of solutions of systems
of second order PDEs via the construction of \emph{shape maps}. The trace of
such maps represents a congruence volume whose collapse we study through an
appropriate evolution equation, akin to Raychaudhuri's equation. We develop the
necessary geometric framework on a suitable jet space in which the shape maps
appear naturally associated with certain linear connections. Explicit
computations are given, along with a nontrivial example
Boundary fluxes for non-local diffusion
We study a nonlocal diffusion operator in a bounded smooth domain prescribing
the flux through the boundary. This problem may be seen as a generalization of
the usual Neumann problem for the heat equation. First, we prove existence,
uniqueness and a comparison principle. Next, we study the behavior of solutions
for some prescribed boundary data including blowing up ones. Finally, we look
at a nonlinear flux boundary condition
Lower and upper bounds for the first eigenvalue of nonlocal diffusion problems in the whole space
We find lower and upper bounds for the first eigenvalue of a nonlocal
diffusion operator of the form T(u) = - \int_{\rr^d} K(x,y) (u(y)-u(x)) \,
dy. Here we consider a kernel where
is a bounded, nonnegative function supported in the unit ball and means a
diffeomorphism on \rr^d. A simple example being a linear function .
The upper and lower bounds that we obtain are given in terms of the Jacobian of
and the integral of . Indeed, in the linear case we
obtain an explicit expression for the first eigenvalue in the whole \rr^d and
it is positive when the the determinant of the matrix is different from
one. As an application of our results, we observe that, when the first
eigenvalue is positive, there is an exponential decay for the solutions to the
associated evolution problem. As a tool to obtain the result, we also study the
behaviour of the principal eigenvalue of the nonlocal Dirichlet problem in the
ball and prove that it converges to the first eigenvalue in the whole
space as
Deformed Double Yangian Structures
Scaling limits when q tends to 1 of the elliptic vertex algebras A_qp(sl(N))
are defined for any N, extending the previously known case of N=2. They realise
deformed, centrally extended double Yangian structures DY_r(sl(N)). As in the
quantum affine algebras U_q(sl(N)), and quantum elliptic affine algebras
A_qp(sl(N)), these algebras contain subalgebras at critical values of the
central charge c=-N-Mr (M integer, 2r=ln p/ln q), which become Abelian when
c=-N or 2r=Nh for h integer. Poisson structures and quantum exchange relations
are derived for their abstract generators.Comment: 16 pages, LaTeX2e Document - packages amsfonts,amssymb,subeqnarra
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